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Solid state lithium ion electrolytes are becoming increasingly important in batteries and related technologies. We have used first-principles modeling techniques based on density functional theory and the nudged elastic band method to examine possible Li ion diffusion mechanisms in idealized crystals of the electrolyte material Li3PO4. In modeling the Li ion vacancy diffusion, we find direct hopping between neighboring meta-stable vacancy configurations to have a minimal migration barrier of Em=0.6 eV. In modeling the Li ion interstitial diffusion, we find an interstitialcy mechanism, involving the concerted motion of an interstitial Li ion and a neighboring Li ion of the host lattice, that can result in a migration barrier as low as Em=0.2 eV. The minimal formation energy of a Li ion vacancy-interstitial pair is determined to be Ef=1.6 eV. Assuming the activation energy for intrinsic defects to be given by EA = Em + Ef/2, the calculations find EA=1.0 eV for ionic diffusion in crystalline γ-Li3PO4, in good agreement with reported experimental values of 1.1-1.3 eV.